Why is the maximum bending moment at the center w(l^2) /8?

[chetzread]

Homework Statement

why the maximum bending moment at the center is w(l^2) /8 ?

Homework Equations

The Attempt at a Solution

shouldn't it be = 0 ?
when we take the moment about the center , the reaction force at the left will generate clockwise moment , while the reaction force at the right will generate antoclockwise moment , they will cancel out each other , resulting the moment = 0? Am i right ?[/B]

 

[pongo38]

Homework Statement

View attachment 102859
why the maximum bending moment at the center is w(l^2) /8 ?

Homework Equations

The Attempt at a Solution

shouldn't it be = 0 ?
when we take the moment about the center , the reaction force at the left will generate clockwise moment , while the reaction force at the right will generate antoclockwise moment , they will cancel out each other , resulting the moment = 0? Am i right ?[/B]

This is similar to another post of yours, to which I replied: What definition of Moment are you using? There are two possible definitions, each of which can be derived from the other. The one I prefer is that the bending moment at a section is the algebraic sum of the moments on ONE SIDE of the section. You have to have a sign convention that should yield the same moment, whether you take it to the left of the section, or to the right. That acts as a check on whether you have made a mistake. Another check is to use the alternative definition (the integral of the shear force diagram- which I don't prefer, because of the arbitrary constant.

 

[chetzread]

This is similar to another post of yours, to which I replied: What definition of Moment are you using? There are two possible definitions, each of which can be derived from the other. The one I prefer is that the bending moment at a section is the algebraic sum of the moments on ONE SIDE of the section. You have to have a sign convention that should yield the same moment, whether you take it to the left of the section, or to the right. That acts as a check on whether you have made a mistake. Another check is to use the alternative definition (the integral of the shear force diagram- which I don't prefer, because of the arbitrary constant.

i am using the first one (algebraic sum of the moments on ONE SIDE of the section.

 

[pongo38]

When you 'take moments about a point', you will get a zero answer for a body in static equilibrium. If you take moments on ONE side of the section, you will get a non-zero answer which should be numerically the same as that algebraic sum on the OTHER side of the section - in this case w(l^2) /8. In using sign conventions, don't attribute a positive moment to a clockwise - anticlockwise- action. The important thing is whether the action is having a hogging effect, or a sagging effect. In your beam with udl, taking momets at the beam centre, the effect of the reaction is sagging, and the effect of the udl is hogging (from the point of view of someone standing underneath the beam). Sign conventions are a big problem that have no world-wide agreement, and plenty of misunderstandings are possible. With practice, the finer points will become clearer to you, and you should recognise that engineering theory is not always the same as engineering practice.

Reference https://www.physicsforums.com/threads/maximum-moment-at-the-center.877845/

 

[chetzread]

When you 'take moments about a point', you will get a zero answer for a body in static equilibrium. If you take moments on ONE side of the section, you will get a non-zero answer which should be numerically the same as that algebraic sum on the OTHER side of the section - in this case w(l^2) /8. In using sign conventions, don't attribute a positive moment to a clockwise - anticlockwise- action. The important thing is whether the action is having a hogging effect, or a sagging effect. In your beam with udl, taking momets at the beam centre, the effect of the reaction is sagging, and the effect of the udl is hogging (from the point of view of someone standing underneath the beam). Sign conventions are a big problem that have no world-wide agreement, and plenty of misunderstandings are possible. With practice, the finer points will become clearer to you, and you should recognise that engineering theory is not always the same as engineering practice.

Reference https://www.physicsforums.com/threads/maximum-moment-at-the-center.877845/

to be exact , my idea is anticlockwise moment = clockwise moment...So total moment =0 , the object in equilibrium...

in this thread
https://www.physicsforums.com/threads/moment-of-beam.877859/#post-5513296
I think it should be EIy" = 0.5Px -P(x-0.5L) -0.5P(L-x) ,that's why ihave the additional -0.5P(L-x) , is it true?

 

[chetzread]

When you 'take moments about a point', you will get a zero answer for a body in static equilibrium. If you take moments on ONE side of the section, you will get a non-zero answer which should be numerically the same as that algebraic sum on the OTHER side of the section - in this case w(l^2) /8. In using sign conventions, don't attribute a positive moment to a clockwise - anticlockwise- action. The important thing is whether the action is having a hogging effect, or a sagging effect. In your beam with udl, taking momets at the beam centre, the effect of the reaction is sagging, and the effect of the udl is hogging (from the point of view of someone standing underneath the beam). Sign conventions are a big problem that have no world-wide agreement, and plenty of misunderstandings are possible. With practice, the finer points will become clearer to you, and you should recognise that engineering theory is not always the same as engineering practice.

Reference https://www.physicsforums.com/threads/maximum-moment-at-the-center.877845/

why can we only consider moment about a point at one side only?

 

[pongo38]

That is the definition of bending moment. It's useful.